Virtual $3$-manifolds

We generalize the class if all compact $3$-manifolds to a class of new objects called virtual $3$-manifolds. Each virtual $3$-manifold determines a $3$-manifold with singularities of the type $\mathrm{Con}(RP^2)$ and may be presented by a triangulation as well as by a special spine. Many properties and invariants of $3$-manifolds can be extended to the virtual ones. We restrict ourselves to mentioning Turaev–Viro invariants and two-sheeted branched coverings of virtual $3$-manifolds.